To determine the equation of a line that passes through the point [tex]\((0,2)\)[/tex] and is perpendicular to the line given by the equation [tex]\(y = \frac{1}{4}x + 5\)[/tex], we can proceed with the following steps:
1. Identify the slope of the given line:
The given line equation is [tex]\(y = \frac{1}{4}x + 5\)[/tex]. From this equation, we can see that the slope ([tex]\(m\)[/tex]) of the given line is [tex]\(\frac{1}{4}\)[/tex].
2. Determine the slope of the perpendicular line:
For two lines to be perpendicular, the product of their slopes must be [tex]\(-1\)[/tex]. This means that if one line has a slope of [tex]\(m\)[/tex], the perpendicular line will have a slope of [tex]\(-\frac{1}{m}\)[/tex].
Given the slope of the original line as [tex]\(\frac{1}{4}\)[/tex], the slope [tex]\(m_{\perp}\)[/tex] of the perpendicular line is:
[tex]\[
m_{\perp} = -\frac{1}{\left(\frac{1}{4}\right)} = -4
\][/tex]
3. Use the point-slope form to write the equation of the line:
The point-slope form of the equation of a line is [tex]\(y - y_1 = m_{\perp}(x - x_1)\)[/tex], where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\(m_{\perp}\)[/tex] is the slope.
Here, the line passes through the point [tex]\((0, 2)\)[/tex], and we have already determined that the slope of our line is [tex]\(-4\)[/tex]. Substituting these values into the point-slope form, we get:
[tex]\[
y - 2 = -4(x - 0)
\][/tex]
Simplify the equation to obtain the slope-intercept form ([tex]\(y = mx + b\)[/tex]):
[tex]\[
y - 2 = -4x
\][/tex]
Add 2 to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[
y = -4x + 2
\][/tex]
Therefore, the equation of the line that is perpendicular to [tex]\(y = \frac{1}{4}x + 5\)[/tex] and passes through the point [tex]\((0, 2)\)[/tex] is:
[tex]\[
y = -4x + 2
\][/tex]
